Page 109 - Phase Space Optics Fundamentals and Applications

90 Chapter Three

parameters, defined by the matrices U 1 , U 2 , and U 3 . Some of the pa-
rameters can also play the role of variables. The choice of the parame-
ters and the number of variables of the GC depends on the particular
application. Thus, if we are interested in the improvement of image
quality or in its manipulation for some feature extraction (e.g., edge
enhancement or image deblurring), then we have to choose U 3 = U −1
1
to represent the result of filtering in the position domain.
A typical optical scheme for GC, GC f,h (U 1 , U 2 , U 3 , r), is a straight-
9
forward generalization of the Van der Lugt processor and consists
of a cascade of two first-order systems described by the matrices U 1
and U 3 (the flexible schemes for the phase-space rotators were con-
sidered in Sec. 3.6) with a diffraction/reflection screen between them
corresponding to multiplication of the passing/reflecting beam by
R [h(·)]. Then with f (·) in the input of this system, we have its GC
U 2
with h(·) at the output plane. The common convolution operation
C f,h (r), Eg. (3.82), arises when U 1 and U 2 correspond to the direct FT
matrices and U 3 to the inverse one. In optical realization, U 3 is usually
the direct FT, and then we have C f,h (−r) at the output plane.

3.7.2 Pattern Recognition
The correlation operation Cor f,h (r) is a measure of the similarity be-
tween two signals f and h. The mathematical verification of this state-
ment is related to the inequality of Schwarz, which permits one to
discriminate two signals of equal energy, since in this case the au-
tocorrelation peak |Cor f, f (0)| is larger than the cross-correlation one

Cor f,h (0) . Note that |Cor f, f (0)| has a maximum in the origin of the
coordinates r = 0. Then by applying the appropriate threshold to
the correlation map |Cor f,h (r)|, the pattern associated with h can be
found on the investigated scene f . Moreover, because the correlation
is shift-invariant, Cor f (r i −v),h(r i ) (r) = Cor f (r i ),h(r i ) (r − v), the positions
of all patterns h, if there are several, can be localized. This operation
−1 ∗
is also performed by the Van der Lugt processor using F [h (·)](u)
as a filter mask.
LetusconsiderasanexampleasetofnumberspresentedinFig.3.8a.
Theamplitudeofthenumericallysimulatedcross-correlationbetween
this image and the reference one (Fig. 3.8b), is given in Fig. 3.8c. The
largest peaks are observed at the positions where 0 is written, which
permits its localization. Note that the value of the cross-correlation
peaks depends on the similarity between 0 and other numbers. Thus,
a relatively large peak is also observed in the end of the middle line
where8iswritten.Moresophisticatedfiltersareusuallyusedforbetter
object discrimination.
If the pattern has to be detected only in a certain region of the scene,
then we must apply the fractional FT convolution, 4,15,19,48 Eq. (3.85),